In the last two decades Kohn-Sham density functional theory (KS-DFT) has become the most widely used electronic structure method in chemistry. This success stems from the low computational cost of the method in comparison with

Whilst progress has been made in developing improved functionals, a number of deficiencies still remain, which lead to problems describing specific types of bonding, bond dissociation and weak interactions. As linear scaling implementations open up the possibility to perform DFT calculations for many 1000’s of atoms the need to remedy these weaknesses is becoming increasingly clear.

My research focusses on various aspects of these exchange-correlation functionals and the details of their evaluations. In particular I am interested in the prospects for using the very robust, accurate and systematic

A central theme to our research is the investigation of ways in which highly accurate systematic wave function based methods such as Moller-Plesset perturbation, coupled cluster, configuration interaction, and multi-reference self-consistent field theories can be used to shed light on the shortcomings and successes of density-functional theory. The ultimate aim being to use this information to improve the approximate exchange-correlation functionals used in practical density-functional calculations.

To explore the connections between density-functional and wave function based methodologies several routes may be explored. In a recent paper we compared the impact of correlation on electronic densities (see above) and Kohn-Sham potentials (see left) derived from typical approximate forms and much more accurate wave function techniques. The study highlighted the idea that Kohn-Sham exchange and correlation functionals in practical use often have counter balancing errors and that these occur even in simple atomic calculations.

In this study Kohn-Sham potentials were derived corresponding to accurate electronic densities calculated using wave function based procedures. These constrained-search based methods provide a route from a given density to the key quantities that an approximate exchange-correlation functional must reproduce. This provides a useful check on the quality of developed approximations but no direct route to new forms for the functional itself. To gain further information about the form of the exchange-correlation functional the adiabatic connection between the model non-interacting Kohn-Sham system and its interacting counterpart can be studied, as described below.

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M. D. Stromsheim, N. Kumar, S. Coriani, E. Sagvolden, A. M. Teale and T. Helgaker,

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I. Grabowski, A. M. Teale, S. Śmiga, and R. Bartlett,

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A. M. Teale, S. Coriani, T. Helgaker,

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A. M. Teale, S. Coriani, T. Helgaker,

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A. M. Teale, S. Coriani, T. Helgaker,

In order to determine the Kohn-Sham multiplicative potentials corresponding to any choice of density a number of constrained search methods have been developed. In particular I have implemented the methods due to Wu and Yang and Zhao-Morrison and Parr in the DALTON quantum chemistry program. When accurate electronic densities are calculated using ab initio methods and input to this procedure, accurate Kohn-Sham potentials and orbitals can be calculated, often providing valuable insight into the failures of standard approaches. The potential labeled as “accurate” in the plot to the right corresponds to the exchange-correlation component of the Kohn-Sham potential that yields the CCSD(T) electronic density.

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O. B. Lutnæs, A. M. Teale, T. Helgaker, D. J. Tozer, K. Ruud and J. Gauss,

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A. Soncini, A. M. Teale, T. Helgaker, F. De Proft and D. J. Tozer,

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M. J. G. Peach, A. M. Miller, A. M. Teale, and D. J. Tozer,

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A. M. Teale, F. De Proft and D. J. Tozer ,

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M. J. G. Peach, A. M. Teale, and D. J. Tozer,

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O. B. Lutnæs, A. M. Teale, T. Helgaker and D. J. Tozer,

The adiabatic connection (AC) provides the apparatus to study the link between the Kohn-Sham non-interacting system and the physical interacting system considered in wave function theories. We have recently implemented a method for the calculation of this connection, based on Lieb maximization. In addition to providing a wealth of useful energetic information the modelling of these curves can provide a constructive route to new functional forms.

In addition to the standard linear adiabatic connection we are investigating the use of alternative parameterizations, such as those based on the error-function and Gaussian attenuated error-function, which have the potential to open up new perspectives on modelling the AC. In addition related forms may be useful in developing short range density functionals for use in range-separated methods.

In addition to the standard linear adiabatic connection we are investigating the use of alternative parameterizations, such as those based on the error-function and Gaussian attenuated error-function, which have the potential to open up new perspectives on modelling the AC. In addition related forms may be useful in developing short range density functionals for use in range-separated methods.

Schematically the adiabatic connection is shown to the right, on the x-axis the parameter lambda represents the strength of the interactions between electrons. In Kohn-Sham theory a model non-interacting system (lambda=0) is introduced, correpsonding to the left hand edge of the plot. In wave function theories the physical interacting system (lambda=1) is considered. The adiabatic connection provides a smooth link between the model KS system and its physical counterpart. Integrating the resulting function (shown schmatically as the grey area) results in a value for the exchange-correlation energy.

These choices of attenuated interactions are often combined with complementary exchange-correlation functionals in range-separated approaches. The way in which these operators emphasize the range of the electronic interactions is here shown by the way in which the curves bunch towards the right hand (short-ranged) end of the plots. The effect is more pronounced for the Gaussian-damped error-function form, which more sharply divides the short and long-ranged interactions.

To the right the corresponding plots for the H2 molecule at the internuclear distances R = 0.7, 1.4, 3.0, 5.0, 7.0 and 10.0 a.u. are shown. In the top plot the usual linear damping of the electronic interactions is used. The curves at short and equilibrium geometries are qualitatively similar to those of the helium atom. However, as the bond is stretched and static correlation becomes more important the curves bend much more sharply. The steepness of these curves can be related to the order of perturbation theory required for their description.

In the middle plot the error-function attenuated interaction is used. Here the plots spread across all lambda values, in contrast to the He iso-electronic series plots above. As the bond is stretched the curves are dominated by the lower lambda (long-ranged) contributions, but contributions from all ranges remain. In the lower plot the Gaussian-damped interaction leads to a much more complicated AC, emphasizing the point that static correlation cannot be simply regarded as purely long-ranged.

To the right the corresponding plots for the H2 molecule at the internuclear distances R = 0.7, 1.4, 3.0, 5.0, 7.0 and 10.0 a.u. are shown. In the top plot the usual linear damping of the electronic interactions is used. The curves at short and equilibrium geometries are qualitatively similar to those of the helium atom. However, as the bond is stretched and static correlation becomes more important the curves bend much more sharply. The steepness of these curves can be related to the order of perturbation theory required for their description.

In the middle plot the error-function attenuated interaction is used. Here the plots spread across all lambda values, in contrast to the He iso-electronic series plots above. As the bond is stretched the curves are dominated by the lower lambda (long-ranged) contributions, but contributions from all ranges remain. In the lower plot the Gaussian-damped interaction leads to a much more complicated AC, emphasizing the point that static correlation cannot be simply regarded as purely long-ranged.

On the y-axis the electronic interaction energies are given. These may correspond to the sum of Classical Coulomb, exhcange and correlation terms (as conventionally divided in KS theory) or any combination of these. To the left the AC curves for the helium iso-electronic series are presented. The curves here represent the correlation interaction energy only. The nuclear charge increases from 1 to 10 for each function with 1 being the most positive line at lambda=1 and 10 being the most negative. The detailed shape of the curves reflects the nature of the correlation in the system under study. Here we see that the H- ion has quite different behaviour to the other members of the series, shown as the much more curved red line.

In addition to depending on the electronic structure of the system under study the AC curves also depend on how the electronic interactions are switched on. We are free to switch on the electronic interactions in any manner that varies smoothly between the non-interacting and fully interacting limits. This corresponds to choosing different ways to modulate the two electron interactions in the electronic Hamiltonian. The plot in the middle on the left is also for the helium iso-electronic series but now corresponds to modulation of the interactions by the error-function. Similarly the lower plot on the left corresponds to a modulation by the error function damped with a Gaussian.

In addition to depending on the electronic structure of the system under study the AC curves also depend on how the electronic interactions are switched on. We are free to switch on the electronic interactions in any manner that varies smoothly between the non-interacting and fully interacting limits. This corresponds to choosing different ways to modulate the two electron interactions in the electronic Hamiltonian. The plot in the middle on the left is also for the helium iso-electronic series but now corresponds to modulation of the interactions by the error-function. Similarly the lower plot on the left corresponds to a modulation by the error function damped with a Gaussian.

The AC curves which are calculated are useful for analyzing the behaviour of the exchange-correlation functional directly and can be compared with standard approximate forms. It is also possible to examine the connections according to the divisions used in wave function based theories. For example below are plots of the AC for the water molecule. On the left the AC curves at CCSD level are presented. On the right the difference between CCSD and CCSD(T) curves are presented, indicating the influence of the perturbative triples correction on the correlation energy of this simple molecule. The behaviour of the triples contribution is captured by a simple quadratic model. For further details on our attempts to model to the AC and the methodology used for this see the publications below.

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M. D. Stromsheim, N. Kumar, S. Coriani, E. Sagvolden, A. M. Teale and T. Helgaker,

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A. M. Teale, S. Coriani, T. Helgaker,

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A. M. Teale, S. Coriani, T. Helgaker,

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A. M. Teale, S. Coriani, T. Helgaker,

Recently an ERC funded project named ABACUS, has begun to extend the functionality of our code to calculate adiabatic connection curves, with the long term goal of developing new exchange-correlation functionals in the context of standard density-functional theory and in addition in current density-functional theory for the application of DFT to molecules in magnetic fields. The main goals of the project are:

A wiki page for this project is available here (login required).

- Implementation of a tool box to Perform Calculations of the AC using a range of electronic structure methods
- Extension of this tool box to cover not only conventional Kohn–Sham DFT but also its current dependent variant, allowing the study of molecules in magnetic fields
- Calculation of adiabatic connection curves for a wide range of systems both with and without applied electromagnetic fields
- Investigations of novel AC paths, for example, emphasizing the range of electronic interactions, of relevance to the development of range-separated hybrid approaches
- Development of new models of the AC based on this data, leading directly to new approximate exchange–correlation functional forms
- Using this data to understand the successes and failures of present-day density functional approximations
- Implementation of current-dependent Kohn–Sham Density Functional Theory using London atomic orbitals in the London program package
- Implementation of new density functional approximations in both the London and Dalton programs

A wiki page for this project is available here (login required).

The dispersion interaction is of fundamental importance to the correct description of the electronic structure of a wide variety of systems in a range of fields. Important examples are molecular biology, where stacking interactions and protein folding are strongly influenced by dispersion interactions; supramolecular chemistry, where the spatial organization of the constituent molecular systems is determined by weak non-covalent interactions; materials chemistry, where the crystallographic structure is often determined by non- covalent interactions; catalytic chemistry, where adsorption of compounds and their propagation on surfaces are strongly influenced by weak interactions; and molecular chemistry and chemical reactivity, where the interactions (both inter- and intra-molecular) of distant molecular fragments are often determined by dispersion. Given the broad range of scale encompassed by these areas of research, Kohn–Sham (KS) density-functional theory (DFT) often represents the method of choice for electronic-structure calculations - particularly, for large and extended systems, where its attractive balance between computational simplicity and reasonable accuracy is essential. It is therefore disturbing that the vast majority of approximate exchange-correlation functionals employed in KS-DFT are incapable of describing the dispersion interaction, although not surprising in the light of their locality (dependence on the density at a point in space and its immediate vicinity only). In many of these fields, the lack of a good description of the dispersion interaction constitutes a serious impediment to the application of the KS-DFT methodology.

Recently we have analyzed the dispersion interaction for the very simplest case of the helium dimer using the adiabatic connection. Our calculations highlight the subtle nature of the dispersion interaction and the challenges faced by density functional approximations which attempt to describe it. In particular errors in the exchange term may overwhelm any reasonable description of the dispersion effect by the correlation term.

By calculating interaction energy ACs and choosing the error-function attenuated operators the resulting interaction ACs become almost entirely long ranged, which supports the success of range-separated DFT / WFT hybrid approaches in this context.

By calculating interaction energy ACs and choosing the error-function attenuated operators the resulting interaction ACs become almost entirely long ranged, which supports the success of range-separated DFT / WFT hybrid approaches in this context.

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M. D. Stromsheim, N. Kumar, S. Coriani, E. Sagvolden, A. M. Teale and T. Helgaker,

The optimized effective potential (OEP) method provides the rigorous way to evaluate orbital dependent exchange-correlation functionals within the Kohn-Sham framework. A local multiplicative potential is determined by minimizing the orbital dependent energy expression. I have implemented the Yang-Wu method for calculating OEPs in the DALTON quantum chemistry program, allowing OEP evaluations of all the functionals available in the program.

The improved orbital energy spectrum arising in these methods can lead to improvements in response properties, as discussed below. In addition, the determination of multiplicative exchange-correlation potentials facilitates comparisons between “pure” density functionals and orbital dependent density functionals, as shown in the figure above right for the CO molecule. The improvement in the asymptotic potential from PBE to B3LYP to O-CAM-B3LYP to the accurate -1/r behaviour is striking. Although significant room for improvement in the structure of the potentials remains.

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I. Grabowski, A. M. Teale, S. Śmiga, and R. Bartlett,

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W. Klopper, A. M. Teale, S. Coriani, T. B. Pedersen and T. Helgaker,

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M. J. G. Peach, J. A. Kattirtzi, A. M. Teale and D. J. Tozer,

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O. B. Lutnæs, A. M. Teale, T. Helgaker, D. J. Tozer, K. Ruud and J. Gauss,

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A. Soncini, A. M. Teale, T. Helgaker, F. De Proft and D. J. Tozer,

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A. M. Teale, A. J. Cohen and D. J. Tozer,

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O. B. Lutnæs, A. M. Teale, T. Helgaker and D. J. Tozer,

One area in which KS-DFT has been unable to compete with ab inito methods is the determination of magnetic response properties. We have investigated the application of the procedures above to the determination of a variety of magnetic response properties: NMR shielding constants, magnetizabilities, spin-rotation constants and rotational g tensors. The use of the OEP method described above can offer consistent improvements in the evaluation of magnetic properties but substantial room for improvement in the underlying functionals remains.

Recently, we have established a set of benchmark data calculated at the CCSD(T) level, taking into account vibrational effects, using London atomic orbitals and extrapolating to the basis set limit, in order to provide a useful reference against which to measure the quality of DFT functionals. The mean relative errors for a variety of functionals against this benchmark data (taking into account vibrational corrections) are illustrated below. The functionals are categorized according to the information on which they are based (the density only, the density and its gradient, the density its gradient and the occupied orbitals). We are presently working towards extending the range of properties that can be calculated using these methods, including electronic response properties.

Recently, we have established a set of benchmark data calculated at the CCSD(T) level, taking into account vibrational effects, using London atomic orbitals and extrapolating to the basis set limit, in order to provide a useful reference against which to measure the quality of DFT functionals. The mean relative errors for a variety of functionals against this benchmark data (taking into account vibrational corrections) are illustrated below. The functionals are categorized according to the information on which they are based (the density only, the density and its gradient, the density its gradient and the occupied orbitals). We are presently working towards extending the range of properties that can be calculated using these methods, including electronic response properties.

Recently, we have established a set of benchmark data calculated at the CCSD(T) level, taking into account vibrational effects, using London atomic orbitals and extrapolating to the basis set limit, in order to provide a useful reference against which to measure the quality of DFT functionals. The mean relative errors for a variety of functionals against this benchmark data (taking into account vibrational corrections) are illustrated on the right. The functionals are categorized according to the information on which they are based (the density only, the density and its gradient, the density its gradient and the occupied orbitals).

The upper plot represents the mean relative errors in rotational g-tensors and the lower in magentizabilities. In each case the improvement in moving from conventional DFT evaluations to OEP based ones is clear, though the accuracy of even the best DFT approaches still lags behind that of CCSD. We are presently working towards extending the range of properties that can be calculated using the OEP-based methods.

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M. J. G. Peach, J. A. Kattirtzi, A. M. Teale and D. J. Tozer,

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O. B. Lutnæs, A. M. Teale, T. Helgaker, D. J. Tozer, K. Ruud and J. Gauss,

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A. Soncini, A. M. Teale, T. Helgaker, F. De Proft and D. J. Tozer,

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A. M. Teale, A. J. Cohen and D. J. Tozer,

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O. B. Lutnæs, A. M. Teale, T. Helgaker and D. J. Tozer,